Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-5x-4y &= 3 \\ -3x-2y &= -1\end{align*}$
Explanation: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $2$ $\begin{align*}5x+4y &= -3\\ -6x-4y &= -2\end{align*}$ Add the top and bottom equations. $-x = -5$ Divide both sides by $-1$ and reduce as necessary. $x = 5$ Substitute $5$ for $x$ in the top equation. $-5( 5)-4y = 3$ $-25-4y = 3$ $-4y = 28$ $y = -7$ The solution is $\enspace x = 5, \enspace y = -7$.